Use Gauss's law to find the charge enclosed by the cube with vertices (±1,±1,±1) if the electric field is E(x,y,z)=5xi+1yj+2zk

Q: 4. Let D = y²z°ãx + 2xyz³ ày + 3xy?z² đz pC/m² in free space. a. Find the total electric flux…

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Q: Find the flux F(x, y, z) = (xy²,x²y, ln(x + y)) directed outwards through the surface of the solid…

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Q: Use Green's Theorem to evaluate the line integral |v - x)dx + (2x – y)dy for the given path. C: x =…

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Q: Verify Green's Theorem by evaluating both integrals y2 dx + x² dy = dA ду for the given path. C:…

A: In this question we can verify the Green's Theorem . i.e, ∫C M dx +Ndy = ∫∫R ∂N∂x-∂M∂ydA .

Q: Evaluate the triple integral ∫∫∫T (x^2) dV where T is the tetrahedron with vertices (0, 0, 0), (1,…

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Q: Find the potential u(x, y) for u +u =0 in the rectangle 0<x< 2 and 0 < y s 2, if the upper side is…

A: The given problem is to solve the potential u(x,y) Laplace partial differential equation, we have to…

Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann equations if ux = v, and u,…

A: Given u = x2-y2v = 2xy Then ux = 2xuy = -2yvx = 2yvy = 2x

Q: If F= represents the flow of a liquid through a parabolic cylinder S: y=x, 0sxs2, 0szsl. That is…

A: If F→=y2,-x2,z2 represents the flow of a liquid through a parabolic cylinder S:y=x2,0≤x≤2,0≤z≤1.…

Q: Use Green's Theorem to find the work done by F(x,y)=x(3x+y)i+xy^2j in moving a particle from the…

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Q: The outward flux of F(x, y, z) = xi + yj + zk across any unit cube is.

A: Given flux is F=xi+yj+zk. Compute div F. divF=∂∂xx+∂∂yy+∂∂zz=1+1+1=3

Q: Evaluate the triple integral∭E xy dV where E is the solid tetrahedon with vertices…

A: Given Data : Triple Integral ∬∫ExydV and E is the solid tetrahedron with vertices 0,0,0,…

Q: 30. Compute (i × j) × j and i × (j × j). What can you conclude about the associativity of the cross…

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Q: 2. The task is to determine the flux of B across G, where G is the positively-oriented portion of…

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Q: 1. If F= (y,z,x), the outward flux through a smooth closed surface (a) depends on the surface. (b)…

A: Given, F=y,z,x We know,  Flux by the Gauss divergence theorem. ∬F.ds=∬∫divF.dvF=yi+zj+xk If you…

Q: Use Gauss's Law to find the charge contained in the solid hemisphere  x2 + y2 + z2 ≤ a2, z ≥ 0,  if…

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Q: 4. Find the work done by the force field F = (x? sin(y),y) on a particle that moves along the cubic…

A: Given: The force field, F→=x2siny,y on a particle that moves along the cubic y=x3 from 1,1 to 2,8.

Q: Use V(P) = k / Kx, y, z2) ds Dp(x, y, z) where k = 8.99 x 10° N - m²/C°, to compute the electric…

A: Consider the provided question, VP=k∫C∂x,y,zDpx,y,zdswhere k = 8.99 x 109 N·m2c2. To compute the…

Q: Study the analytically of w = L_r(z) at the points z = 0, z = v2,z = -2i

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Q: Calculate the double integral (x + y)²ex* - y² dx dy where R is the square with vertices (2, 0), (0,…

A: According to the given information, it is required to find the integral over R where R is square.

Q: find the outward flux of F across the boundary of D.  F = 2xyi + 2yzj + 2xzk D: The cube cut from…

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Q: Use Gauss's Law to find the charge contained in the solid hemisphere x2 + y2 + z² < a², z z 0, if…

A: Given:The Electric field, E→=xi⇀+yj⇀+7zk⇀Equation of solid hemisphere isx2+y2+z2≤a2       ,z≥0To…

Q: For this problem why did we integrated from -90degre to 90degree, instead of integrating from…

A: To discuss the suggested solution for the length of the given curve

Q: Solve Laplace's equation u +u, = 0, 0<x< 5, 0 < y< 4_for a rectangular plate %3D subject to the…

A: Laplace equation: uxx+uyy=0 For a rectangular plate 0≤x≤a, 0≤y≤b With boundary conditions: u(0,y)=0,…

Q: Use Green's Theorem to evaluate the line integral S(x² + y²)dx + (x² – y²) dy where C is the…

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Q: Evaluate the triple integral. 6xyz dV, where T is the solid tetrahedron with vertices (0, 0,0), (1,…

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Q: Use the Chain Rule to find dz/əs and dz/at. z = tan(u/v), u = 8s + 3t, V = 3s 8t az as az Ət II

A: Given data: *z=tan(uv) *u=8s+3t and v=3s-8t The partial derivative of z with respect to s is given…

Q: tor field F = 21 + 4j + 3k across the surface S.

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Q: Use 6(x, y, z) ds Dp(x, y, z) V(P) = k where k = 8.99 × 10° N m²/C2, to compute the electric…

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Q: Evaluate the triple Integral. 3xyz dV, where T

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Q: Find the counterclockwise circulation and the outward flux of thefield F = (-sin y)i + (x cos y)j…

A: Use tangential form of Green's Theorem to find the counterclockwise circulation. Counterclockwise…

Q: Use Gauss's law to find the charge enclosed by the cube with vertices (±2, ±2, ±2) if the electric…

A: The net charge enclosed by a closed surface S is given by Q=ε0∫∫s E.dS We can divide S into 6…

Q: Identify the work done, J. F.dr by an electric field, E = (x + xy)î + (y + 5)f – 5zk V/m to move an…

A: Work done along the curve C by vector field F . Where C is given by parametric equation r(t)…

Q: 4) Use Stokes theorem to evaluate SF.dr, where с F(x, y, z) = <.xy, yz, *x), where C is the triangle…

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Q: Evaluate the triple integral II. x* dV, where T' is the solid tetrahedron h vertices (0, 0, 0),…

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Q: Solve the sifferential equation using tHce Laplace transforms- you y =I y lo)=y'(0) = y"(0)= 6 %3D

A: The Laplace transform of the function f(t) is represented as F(s) or L[f(t)], it is defined as…

Q: Determine if F=(e* cos y + yz)i + (xz-e* siny) j+(xy+z)k is conservative. If it is conservative,…

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Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann equations if ux = v, and u,…

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Q: The strength E of an electric field at a point (x, y, z) due to an infinitely long charged wire…

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Q: Find the work done by F in moving a particle once counterclockwise around the given curve = (2x –…

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Q: Verify Stokes' Theorum for F(x,y,z) = (y,-x,yz) if S is the paraboloid z = x2 + y2 with x2 + y2 = 1,…

A: Given  Fx,y,z=y,-x,yz Then, F→=yi^-xj^+yzk^

Q: If f(x,y)=x2+y², then the gradient field at (1,-1) is given by: O. f(x,y)= -2i + 2j O II. f(x,y)=…

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Q: Find the flux of F=(x, y, z) across the solid that is enclosed by the hemi = 0. Assume an outward or…

A: The Gauss Divergence Theorem states that the volume integral of the divergence of a vector field A…

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Q: A cube is defined by 1 < x, y, z < 1.2. If D = 2x?ya, + 3x?y²a, -6z²a.C/m? (a) Apply Gauss's law to…

A: Consider the provided question, A cube is defined by 1&lt;x, y, z&lt;1.2 Given,…

Q: 3. Consider two charges of Q = 2x10 C at M(2, 1, 3) and a charge of Q =-3×10°C at N(3, 0, 5) in a…

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Q: dy _ у-х dx 1+x y(0) =1 at the point x=0.1

A: Given differential equation is dydx=y-x1+x,y0=1.

Q: 10. Find the work done by the force F=vx +2y y - 7 for a particle that moves between points (-1,0,1)…

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Q: Given that F(x, y, z)=sin x i+cos yj+ xzk. Evaluate , F.dr where C is given by r(t)={r',-f,t) :…

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Q: Use Gauss's law to find the charge enclosed by the cube with vertices (±2, +2, +2) if the electric…

A: The given Electric field: E(x,y,z)=xi+yj+zk The given vertices of cube is (±2,±2,±2) The expression…